A Mathematical Model for Assessing How Obesity-Related Factors Aggravate Diabetes

Abstract

Obesity-related factors have been associated with beta cell dysfunction, potentially leading to Type 2 diabetes. To address this issue, we developed a comprehensive obesity-based diabetes model incorporating fat cells, glucose, insulin, and beta cells. We established the model’s global existence, non-negativity, and boundedness. Additionally, we introduced a delay to examine the effects of impaired insulin production resulting from beta-cell dysfunction. Bifurcation analyses were conducted for delay and non-delay models, exploring the model’s dynamic transitions through backward and forward Hopf bifurcations. Utilizing the maximal Pontryagin principle, we formulated and evaluated an optimal control problem to mitigate diabetic complications by reducing the prevalence of overweight individuals and halting disease progression. Comparative graphical outputs were generated to demonstrate the beneficial effects of glucose-regulating medication and regular exercise in managing diabetes.

Appendix A

$$\begin{aligned} A_{11} =& \dfrac{b G I p}{(1 + G p + F q)^{2}} - \dfrac{b I}{1 + G p + F q} -d_{1}, \\ A_{12} =&-\dfrac{b G}{1 + G p + F q}, \\ A_{13} =&0, \\ A_{14} =& c + \dfrac{b G I q}{(1+ G p + F q)^{2}}, \\ A_{21} =& \dfrac{B \mu}{ G+k_{1}}-\dfrac{B G \mu}{( G +k_{1})^{2}}, \\ A_{22} =& -d_{2}, \\ A_{23} =&\dfrac{G \mu e^{-\eta \tau}}{ G + k_{1}}, \\ A_{24} =& 0, \\ A_{31} =& \dfrac{B h}{G + k_{2}}- \dfrac{B G h}{(G + k_{2})^{2}}, \\ A_{32} =&0, \\ A_{33} =& \dfrac{G h}{G + k_{2}}-d_{3} - \delta F^{2} , \\ A_{34} =& -2 B \delta F, \\ A_{41} =& 0, \\ A_{42} =& 0, \\ A_{43} =& 0, \\ A_{44} =& (1 - \dfrac{F}{k_{3}}) r - \dfrac{F r}{k_{3}}. \end{aligned}$$

$p_{1} = d_{1} + d_{2} + d_{3} + \delta F^{2} - \dfrac{G h}{G + k_{2}} - \dfrac{ b G I p}{v^{2}} + \dfrac{b I}{v} - r + \dfrac{2 F r}{ k_{3}}$, and $v = (1 + G p + F q)$,

$p_{2} = \dfrac{ b G^{2} h I p}{(G + k_{2}) v^{2}}-\dfrac{b d_{3} G I p}{v^{2}} - \dfrac{ b \delta F^{2} G I p}{v^{2}} + \dfrac{b d_{3} I}{ v} + \dfrac{b \delta F^{2} I}{v} - \dfrac{ b G h I}{(G + k_{2}) v} - d_{3} r - \delta F^{2} r + \dfrac{G h r}{G + k_{2}} + \dfrac{2 d_{3} F r}{k_{3}} + \dfrac{2 \delta F^{3} r}{k_{3}} - \dfrac{ 2 F G h r}{(G + k_{2}) k_{3}} + \dfrac{b G I p r}{v^{2}} - \dfrac{ 2 b F G I p r}{k_{3} v^{2}} - \dfrac {b I r}{v} + \dfrac{2 b F I r}{k_{3} v } + d_{1} \left (d_{2} + d_{3} + \delta F^{2} - \dfrac{G h}{G + k_{2}} - r + \dfrac{2 F r}{k_{3}}\right ) + d_{2} \bigg (d_{3} + \delta F^{2} - \dfrac{G h}{G + k_{2}} - \dfrac{b G I p}{v^{2}} + \dfrac{ b I}{v} - r + \dfrac{2 F r}{k_{3}} \bigg )+ \dfrac{ b B G k_{1} \mu}{(G + k_{1})^{2} v}$,

$p_{3}= \dfrac{d_{2} ( d_{3} (G + k_{2}) -G h + \delta F^{2} (G + k_{2})) (2 F - k_{3}) r}{(G + k_{2}) k_{3}} + \dfrac{1}{(G + k_{2}) k_{3}}(d_{1} d_{2} ( d_{3} (G + k_{2}) -G h + \delta F^{2} (G + k_{2})) k_{3} + d_{1} (G (d_{2} + d_{3} + \delta F^{2} - h) + (d_{2} + d_{3} + \delta F^{2}) k_{2}) (2 F - k_{3}) r) + \dfrac{1}{v^{2}} b ( \dfrac{2 d_{3} F I r}{k_{3}}-d_{3} I r + \delta F^{2} G I p r - \dfrac{G^{2} h I p r}{ G + k_{2}} - \dfrac{2 \delta F^{3} G I p r}{k_{3}} + \dfrac{ 2 F G^{2} h I p r}{(G + k_{2}) k_{3}} - d_{3} F I q r + \dfrac{2 d_{3} F^{2} I q r}{ k_{3}} - \delta F^{2} I v r + \dfrac{G h I v r}{ G + k_{2}} + \dfrac{2 \delta F^{3} I v r}{k_{3}} - \dfrac{ 2 F G h I v r}{(G + k_{2}) k_{3}} + \dfrac{ d_{2} I (1 + F q) ( d_{3} (G + k_{2}) k_{3} -G h k_{3} + \delta F^{2} (G + k_{2}) k_{3} + (G + k_{2}) (2 F - k_{3}) r)}{(G + k_{2}) k_{3}} + \dfrac{ B d_{3} G k_{1} \mu}{(G + k_{1})^{2}} + \dfrac{B d_{3} G^{2} k_{1} p \mu}{(G + k_{1})^{2}} + \dfrac{ B d_{3} F G k_{1} q \mu}{(G + k_{1})^{2}} - \dfrac{ B \delta F^{2} G^{2} v \mu}{(G + k_{1})^{2}} + \dfrac{ B \delta F^{2} G v \mu}{G + k_{1}} + \dfrac{ B G^{3} h v \mu}{(G + k_{1})^{2} (G + k_{2})} - \dfrac{ B G^{2} h v \mu}{(G + k_{1}) (G + k_{2})} + \dfrac{ B G^{2} v r \mu}{(G + k_{1})^{2}} - \dfrac{ B G v r \mu}{G + k_{1}} - \dfrac{ 2 B F G^{2} v r \mu}{(G + k_{1})^{2} k_{3}} + \dfrac{ 2 B F G v r \mu}{(G + k_{1}) k_{3}})$,

$p_{4} = \dfrac{( d_{3} (G + k_{2})-G h + \delta F^{2} (G + k_{2})) (2 F - k_{3}) r (b d_{2} I (G + k_{1})^{2} (1 + F q) + d_{1} d_{2} (G + k_{1})^{2} v^{2} + b B G k_{1} v \mu )}{(G + k_{1})^{2} (G + k_{2}) k_{3} v^{2}}$,

$q_{1} = \dfrac{b B G^{2} h k_{2} u}{v(G + k_{1}) (G + k_{2})^{2}}$,

$q_{2} = \dfrac{b B G^{2} h k_{2} (2 F - k_{3}) r \mu}{(G + k_{1}) (G + k_{2})^{2} k_{3} v}$.